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1

Voting to close. People are at the repeating-other-people's-answers stage now.
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Qiaochu YuanAug 21 '10 at 18:23

2

The question has been closed as no longer relevant. It had a long and healthy life, but the large number of answers has become unwieldy. If the question had been asked more recently, it would probably have been closed sooner as being "overly broad". I encourage people who are interested in following up issues raised in the question or the answers with further questions. Please be specific!
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Pete L. ClarkAug 22 '10 at 9:02

@Gerald: I certainly agree. But I think this particular equation is so pretty I once had it henna tattoo-ed on my hand. @Sam: It's just a matter of eye, beholder etc. I don't like minus signs, but do like 0 on the RHS.
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Sonia BalagopalanOct 29 '09 at 15:29

9

I think people ascribe to this formula excessive mystery. The mystery disappears if one replaces the usual exponential with the matrix exponential on M_2(R) and then this is just the statement that the only trajectory of a particle whose velocity is always perpendicular to its displacement (and in the same proportion) is a circle.
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Qiaochu YuanDec 8 '09 at 18:04

32

I think the people who ascribe to this formula excessive mystery might not find matrix exponentials particularly less mysterious.
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Cam McLemanMar 4 '10 at 18:16

I wouldn't call it trivial. There is also a nice combinatorial proof for it.
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Victor ProtsakJun 3 '10 at 14:12

4

I am a bit late in seeing this post, but I completely agree with you, Yaakov. The equality is a bit related to the following pattern which I discovered as a child but continues to amaze me to this day: $1=1^3$; $3+5=2^3$; $7+9+11=3^3$; $13+15+17+19=4^3$;... Many mathematicians know that the sum of the first n odd numbers is n2, but I think very few are aware of this trivial yet incredible pattern. I actually wrote a little post about this on a math blog (the Everything Seminar):
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Peter LuthyNov 3 '10 at 5:34

$$ \frac{24}{7\sqrt{7}} \int_{\pi/3}^{\pi/2} \log \left| \frac{\tan
t+\sqrt{7}}{\tan t-\sqrt{7}}\right|\\ dt = \sum_{n\geq
1} \left(\frac n7\right)\frac{1}{n^2}, $$
where $\left(\frac n7\right)$ denotes the Legendre symbol. Not really
my favorite identity, but it has the interesting feature that it is a
conjecture! It is a rare example of a conjectured explicit identity
between real numbers that can be checked to arbitrary accuracy.
This identity has been verified to over 20,000 decimal places.
See J. M. Borwein and D. H. Bailey, Mathematics by Experiment:
Plausible Reasoning in the 21st Century, A K Peters, Natick, MA,
2004 (pages 90-91).

If A is the adjacency matrix of a finite graph G, then the coefficient of t^k counts the number of non-negative integer linear combination of aperiodic closed walks on G (without a distinguished vertex) with a total of k vertices. This is an equivalent to an Euler product for the RHS which is again straightforward to prove algebraically and very interesting combinatorially.
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Qiaochu YuanOct 29 '09 at 4:24

Both sides as formal power series work out to 1 + z + z^2 + z^3 + ..., where all the coefficients are 1. This is an analytic version of the fact that every positive integer can be written in exactly one way as a sum of distinct powers of two, i. e. that binary expansions are unique.

True, but that's not what's going on here. One can define 0! by extending the rule n!=n*(n-1)! with n=1, or by computing that \Gamma(1) in its integral form is indeed 1. And it's only very mildly a convention that we choose \Gamma as the interpolating function for the factorial function.
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Cam McLemanMar 4 '10 at 18:23

6

Or, 0! is the number of bijective maps on a set with 0 elements, since the empty set is the only such map.
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Jonas MeyerMar 19 '10 at 3:52

3

Ha! Very funny! Reminded me of when one of my professors ended a proof with the statement to the effect that the Q is an element in set D. (get it?)
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BurhanMay 21 '10 at 18:25

It's too hard to pick just one formula, so here's another: the Cauchy-Schwarz inequality:

||x|| ||y|| >= |(x.y)|, with equality iff x&y are parallel.

Simple, yet incredibly useful. It has many nice generalizations (like Holder's inequality), but here's a cute generalization to three vectors in a real inner product space:

||x||2 ||y||2 ||z||2 + 2(x.y)(y.z)(z.x) >= ||x||2(y.z)2 + ||y||2(z.x)2 + ||z||2(x.y)2, with equality iff one of x,y,z is in the span of the others.

There are corresponding inequalities for 4 vectors, 5 vectors, etc., but they get unwieldy after this one. All of the inequalities, including Cauchy-Schwarz, are actually just generalizations of the 1-dimensional inequality:

||x|| >= 0, with equality iff x = 0,

or rather, instantiations of it in the 2nd, 3rd, etc. exterior powers of the vector space.

I think that Weyl's character formula is pretty awesome! It's a generating function for the dimensions of the weight spaces in a finite dimensional irreducible highest weight module of a semisimple Lie algebra.

I didn't say it was hard to prove, but you've asked a fair question. It struck me as beautiful when I first learned as an undergrad of this way to see that there are infinitely many different infinite cardinals.
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Jonas MeyerJan 10 '10 at 18:40

Once one decides that the "right" definition of $n!$ when $n$ isn't a natural number is $\Gamma(n+1)$, this formula becomes equivalent to one of my favorites: $(-{\frac{1}{2}})!=\sqrt{\pi}$.
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Andreas BlassAug 21 '10 at 18:55

For X a based smooth manifold, the category of finite covers over X is equivalent to the category of actions of the fundamental group of X on based finite sets:

\pi-sets === et/X

The same statement for number fields essentially describes the Galois theory. Now the idea
that those should be somehow unified
was one of the reasons in the development of abstract schemes, a very fruitful topic that is studied in the amazing area of mathematics called the abstract algebraic geometry. Also, note that "actions on sets" is very close to "representations on vector spaces" and this moves us in the direction of representation theory.

Now you see, this simple line actually somehow relates number theory and representation theory. How exactly? Well, if I knew, I would write about that, but I'm just starting to learn about those things.

(Of course, one of the specific relations hinted here should be the Langlands conjectures, since we're so close to having L-functions and representations here!)

where j(q)-744 = \sumn >-2 c(n) qn = q-1 + 196884q + 21493760q2 + ... The left side is a difference of power series pure in p and q, so all of the mixed terms on the right cancel out. This yields infinitely many identities relating the coefficients of j.

The formula $\displaystyle \int_{-\infty}^{\infty} \frac{\cos(x)}{x^2+1} dx = \frac{\pi}{e}$. It is astounding in that we can retrieve $e$ from a formula involving the cosine. It is not surprising if we know the formula $\cos(x)=\frac{e^{ix}+e^{-ix}}{2}$, yet this integral is of a purely real-valued function. It shows how complex analysis actually underlies even the real numbers.

Addendum to $e^{i \pi}$

Benjamin Peirce apparently liked this mathematical synonym for the additive inverse of $1$ so much that he introduced three special symbols for $e, i, \pi$ — ones that enable $e^{i \pi}$ to be written in a single cursive ligature, like so: